This post is in some ways an antithesis of my previous postings on hard and soft analysis. In those posts, the emphasis was on taking a result in soft analysis and converting it into a hard analysis statement (making it more “quantitative” or “effective”); here we shall be focusing on the reverse procedure, in which one harnesses the power of infinitary mathematics – in particular, ultrafilters and nonstandard analysis – to facilitate the proof of finitary statements.

Arguments in hard analysis are notorious for their profusion of “epsilons and deltas”. In the more sophisticated arguments of this type, one can end up having an entire army of epsilons that one needs to manage, in particular choosing each epsilon carefully to be sufficiently small compared to other parameters (including other epsilons), while of course avoiding an impossibly circular situation in which a parameter is ultimately required to be small with respect to itself, which is absurd. This art of epsilon management, once mastered, is not terribly difficult – it basically requires one to mentally keep track of which quantities are “small”, “very small”, “very very small”, and so forth – but when these arguments get particularly lengthy, then epsilon management can get rather tedious, and also has the effect of making these arguments unpleasant to read. In particular, any given assertion in hard analysis usually comes with a number of unsightly quantifiers (For every there exists an N…) which can require some thought for a reader to parse. This is in contrast with soft analysis, in which most of the quantifiers (and the epsilons) can be cleanly concealed via the deployment of some very useful terminology; consider for instance how many quantifiers and epsilons are hidden within, say, the Heine-Borel theorem: “a subset of a Euclidean space is compact if and only if it is closed and bounded”.

For those who practice hard analysis for a living (such as myself), it is natural to wonder if one can somehow “clean up” or “automate” all the epsilon management which one is required to do, and attain levels of elegance and conceptual clarity comparable to those in soft analysis, hopefully without sacrificing too much of the “elementary” or “finitary” nature of hard analysis in the process.

A concrete special case of this theorem relates to the product Cantor set K, consisting of all points (x,y) in the unit square whose base 4 expansion consists only of 0s and 3s. This is a compact one-dimensional set of finite length, which is purely unrectifiable, and so Besicovitch’s theorem tells us that almost every projection of K has measure zero. (One consequence of this, first observed by Kahane, is that one can construct Kakeya sets in the plane of zero measure by connecting line segments between one Cantor set and another.)

Lebesgue density theorem. Let be Lebesgue measurable. Then for almost every , we have as , where |A| denotes the Lebesgue measure of A.

In other words, almost all the points x of A are points of density of A, which roughly speaking means that as one passes to finer and finer scales, the immediate vicinity of x becomes increasingly saturated with A. (Points of density are like robust versions of interior points, thus the Lebesgue density theorem is an assertion that measurable sets are almost like open sets. This is Littlewood’s first principle.) One can also deduce the Lebesgue differentiation theorem back from the Lebesgue density theorem by approximating f by a finite linear combination of indicator functions; we leave this as an exercise.

In the field of analysis, it is common to make a distinction between “hard”, “quantitative”, or “finitary” analysis on one hand, and “soft”, “qualitative”, or “infinitary” analysis on the other. “Hard analysis” is mostly concerned with finite quantities (e.g. the cardinality of finite sets, the measure of bounded sets, the value of convergent integrals, the norm of finite-dimensional vectors, etc.) and their quantitative properties (in particular, upper and lower bounds). “Soft analysis”, on the other hand, tends to deal with more infinitary objects (e.g. sequences, measurable sets and functions, -algebras, Banach spaces, etc.) and their qualitative properties (convergence, boundedness, integrability, completeness, compactness, etc.). To put it more symbolically, hard analysis is the mathematics of , , , and [1]; soft analysis is the mathematics of 0, , , and .

At first glance, the two types of analysis look very different; they deal with different types of objects, ask different types of questions, and seem to use different techniques in their proofs. They even use[2] different axioms of mathematics; the axiom of infinity, the axiom of choice, and the Dedekind completeness axiom for the real numbers are often invoked in soft analysis, but rarely in hard analysis. (As a consequence, there are occasionally some finitary results that can be proven easily by soft analysis but are in fact impossible to prove via hard analysis methods; the Paris-Harrington theorem gives a famous example.) Because of all these differences, it is common for analysts to specialise in only one of the two types of analysis. For instance, as a general rule (and with notable exceptions), discrete mathematicians, computer scientists, real-variable harmonic analysts, and analytic number theorists tend to rely on “hard analysis” tools, whereas functional analysts, operator algebraists, abstract harmonic analysts, and ergodic theorists tend to rely on “soft analysis” tools. (PDE is an interesting intermediate case in which both types of analysis are popular and useful, though many practitioners of PDE still prefer to primarily use just one of the two types. Another interesting transition occurs on the interface between point-set topology, which largely uses soft analysis, and metric geometry, which largely uses hard analysis. Also, the ineffective bounds which crop up from time to time in analytic number theory are a sort of hybrid of hard and soft analysis. Finally, there are examples of evolution of a field from soft analysis to hard (e.g. Banach space geometry) or vice versa (e.g. recent developments in extremal combinatorics, particularly in relation to the regularity lemma).)

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